where $x > 0$ and both $c$ and $d=e^{c}$ are integration constants.
So the general solution consists of all the functions defined (in implicit form) by the parametric family of equations:
\begin{equation} \label{implgensol}
x^{y} = d sinx
\end{equation}
where $x > 0$.(b). In order to determine the special solution coming through $(0,1)$ we have to substitute $x=0$, $y=1$ in the general solution \eqref{implgensol}, getting: $ \ \ \ 0=d sin0$.
Thus, all solutions \eqref{implgensol}, for any $d \in \mathbb{R}$ are coming through $(0,1)$.